[/quote]
,
such that 
.
students, each pair of students have watched a movie together. Let the set of movies watched by a student be his movie collection. If every student has watched every movie at most once, at least how many different movie collections can these students have?
be non-negative real numbers such that 
Prove that
and 
.
and 
be real numbers. Let 
be the matrix with entries ![\[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\]](http://latex.artofproblemsolving.com/c/0/8/c0856fe7b2b113180659952c4d216a01d8849f64.png)
Suppose that 
is an 
matrix with entries 
,
such that the sum of the elements in each row and each column of 
.
.
be an acute triangle and let 
.
When does equality hold?
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
![\[
3 + (2 - \sqrt{3}) \cdot \frac{(b-c)^2}{b+(\sqrt{3}-1)c} \leq a+b+c
\]](http://latex.artofproblemsolving.com/5/a/8/5a83eaa7470bcca6546599ff4e296e4efec69b16.png)
and determine all the cases when the equality occurs.
have the same incenter 
.
is concyclic iff 
is concurrent
be the nonnegative integers in 
where 
.
.
.
such that 
where 
,
?
where 
,
be a scalene triangle with circumcircle 
and circumcircle 
.
and 
as the midpoints of 
and 
,
to 
.
meets 
,
intersects 
.
at 
and 
at 
.
intersect 
,
intersect 
.
is the reflection of 
and 
lies on the circumcircle of 
.
and 
and determine the equality cases.
Y by Adventure10, Mango247
Suppose 
is an inner point in a convex pentagon 
such that
![\[ \angle APB = \angle BPC = \angle CPD = \angle DPE = \angle EPA \]](//latex.artofproblemsolving.com/9/1/0/9104046161a1e431a3323ac34a19439679bbfbe4.png)
and all 5 -excircles of 
have the same radius.
Prove that is a regular polygon.
(By definition, a
pentagon is a pentagon where all of its angles and sides are equal).
is an inner point in a convex pentagon 
such that![\[ \angle APB = \angle BPC = \angle CPD = \angle DPE = \angle EPA \]](http://latex.artofproblemsolving.com/9/1/0/9104046161a1e431a3323ac34a19439679bbfbe4.png)
and all 5 -excircles of 
have the same radius.Prove that
is a regular polygon.(By definition, a
pentagon is a pentagon where all of its angles and sides are equal).This post has been edited 4 times. Last edited by IndoMathXdZ, Dec 24, 2018, 6:01 AM
![\[PA + PB + AB = PB + PC + BC = \ldots = PE + PA + EA\]](http://latex.artofproblemsolving.com/6/1/e/61ec6c1a352f630d44a66a352b66678c411b24e7.png)
and ![\[\frac{PA^2 + PB^2 - AB^2}{PA\cdot PB} = \frac{PB^2+PC^2-BC^2}{PB\cdot PC} = \ldots\]](http://latex.artofproblemsolving.com/7/1/0/7107e5bb26fdc97a73400b04d5771a7bf792820a.png)
which gives the result (hopefully 
)
as 
respectively. Also, let us call the excenter of 
by 
.
is the foot of perpendicular from 
to 
,
to 
is the semiperimeter of 
,
.
.
(BA, BC are tangents to symmetric circles from a point on 
)
.
.
.
.
